Non-squareness properties of Orlicz–Lorentz sequence spaces

In this paper criteria for non-squareness properties (non-squareness, local uniform non-squareness and uniform non-squareness) of Orlicz–Lorentz sequence spaces λφ,ω and of their n-dimensional subspaces (n⩾2) as well as of the subspaces (λφ,ω)a of all order continuous elements in λφ,ω are given. Since degenerate Orlicz functions φ and degenerate weight sequences ω are also admitted, these investigations concern the most possible wide class of Orlicz–Lorentz sequence spaces. Finally, as immediate consequences, criteria for all non-squareness properties of Orlicz sequence spaces, which complete the results of Sundaresan (1966) [53], , Hudzik (1985) [23], , Hudzik (1985) [24], , are deduced. It is worth recalling that uniform non-squareness is an important property, because it implies super-reflexivity as well as the fixed point property (see James (1964) [31], , James (1972) [33], and García-Falset et al. (2006) [19]).